Chapter 6: Similarity By Elana, Kate, and Rachel

6.1 Ratios, Proportions, and the Geometric Mean  K HDbase unit D C M  Proportions: equation that states 2 ratios are equal  Cross-Product Property: In a proportion the product of the means equals the product of the extremes  Ex. If = then ad = bc (b≠0 and d≠0)  Geometric Mean: The geometric mean of two positive numbers a and b is the positive number that satisfies  Ex. If =  x 2 = ab  x =

6.1 Examples  Ex. 1 Solve the proportion. 6+h = 8  3(6+h) = 8(4) 18+3h = 32  3h = 14  h = 14/3  Ex. 2 Find the geometric mean of 3 and 27 3 × 27 = 81  = 9 4 3

6.2 Using Properties to Solve Geometry Problems Properties of Proportions: Reciprocal Property: If 2 ratios are equal, then their reciprocals are equal. Ex. If = then = If you interchange the means of a proportion, then you create another true proportion. Ex. If = then = If you add the value of each ratio’s denominator to its numerator you form another true proportion. Ex. =

Scale Drawing: a drawing that is the same shape as the object it represents “SCALE”: a ratio that describes the dimensions in a drawing and how it relates to the actual dimension of the object

6.2 Examples TRY THIS! Find the scale of the rectangles. (blue to pink) 4 cm. 7 cm. 4 cm:7 cm is the scale.

6.3 Use Similar Polygons Similar Polygons: 2 polygons are proportional if corresponding angles are congruent and corresponding side lengths are proportional Pink ~ Purple Scale Factor: the ratio of the lengths of 2 corresponding sides (only in similar polygons)

Theorem 6.1 Perimeters of Similar Polygons: If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths The scale factor of the corresponding sides is ½. The ratio of the perimeters is 16/32, which is also ½. Corresponding Lengths in Similar Polygons: If 2 polygons are similar, then the ratio of any 2 corresponding lengths in the polygons is equal to the scale factor of the similar polygon.

6.4 Prove Triangles Similar by AA Triangle Similarity: 2 triangles are similar if 2 pairs of corresponding angles are congruent AA (Angle-Angle) Similarity Postulate: If 2 angles of one triangle are congruent to 2 angles of another triangle, then the 2 triangles are similar.

Indirect Measurement: useful way to find measurements indirectly is by using similar triangles TRY THIS! The real horse’s tail is 2 ft. long and ½ a foot wide. The scale drawing of the horse’s tail is 5 in long. What is the width of the scale horse’s tail? 2x = 5(.5)  2x = 2.5 X = 1.25 The width of the scale of the horse’s tail is 1.25 in.

6.5 Prove Triangles Similar by SSS and SAS SSS Similarity Theorem: If the corresponding side lengths of 2 triangles are proportional, then the triangles are similar. ABC ~ RST A BC R ST

SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. is congruent to and, then XYZ ~ MNP X YZ M PN

6.5 Examples TRY THIS! Find the value of x that makes triangle RST ~ triangle EFG. R ST E FG 3 x = 2x = 2x x = ½

6.6 Use Proportionality Theorems Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other 2 sides, then it divides the 2 sides proportionally.

Ex. What is the length of segment RT? 18 = 6x x = 3

Converse of Triangle Proportionality Theorem: If a line divides 2 sides of a triangle proportionally then it is parallel to the third side. Ifthen segment DE is parallel to segment BC. Theorem 6.6: If 3 parallel lines intersect 2 transversals, then they divide the transversals proportionally.

Theorem 6.7: If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the 2 other sides.

Try this! Find BC x 54 = 12 x x = 9/2 = 4.5

6.7 Similarity Transformations Dilations: transformation that stretches or shrinks a figure to create a new figure (Dilation is enlarged or reduced about a fixed point called center of dilation) Scale Factor: is the ratio of a side length of the image to its corresponding side length of the original figure ***TIP: scale factor is image to pre-image Scale factor is 6/1.5 = 4/1

Coordinate Notation for Dilations: You can describe a dilation with respect to the origin with (x,y)  (kx,ky) k = scale factor 0 1 (enlargement)

TRY THIS! Draw triangle ABC. A(-2,-2) B(-1,2) C(2, 1) Use the scale factor of 2 to find the image. A’ (-2(2), -2(2))  A’ (-4,-4) B’ (-1(2), 2(2))  B’ (-2, 4) C’ (2(2), 1(2))  C’ (4, 2)